Planning TS Trajectory Using MLAT in $\mathrm{o}(\mathrm{n}\log \mathrm{n})$

Abstract

The Multi-Level Adaptive Technique (MLAT) is a technique used for solving problems approximately and iteratively at levels with various resolutions, and then injecting the corresponding solution at each level into a problem with close resolution for the next iteration. MLAT was originally applied solving successfully partial differential equations using the so called Multi Grid method, using a set of grids with gradually varying mesh sizes. Each grid was treated separately using the so-called relaxation, and then traversing (injection from fine to coarse grid, and interpolation from coarse to fine grid) the data between the grids at two close levels. The Traveling Salesman (TS) problem - popular in Motion Planning, solved by MLAT using Graph Theory. The vertices are specially partitioned into groups, which are moved into more general groups, and again into higher level groups. This grouping is repeated until the number of elements at the highest level collected, does not have too many elements in order to enable their easy and fast manipulation. The TS problem, namely, searching for the shortest path traversing all the vertices in the graph, is approximated by MLAT by partitioning the graph into small subgraphs by selecting the odd vertices in the original graph; each subgraph is similarly divided again into a smaller subgraph. This procedure is repeated until a subgraph is obtained, which is small enough. The TS problem is solved using the coarsest subgraph obtained. This solution is injected into the finer subgraph to improve the approximate solution by relaxation, on the current subgraph. The injection from a coarse graph to a fine graph is followed by relaxation, which is repeated on all the pairs of the graph and its subgraph. Then the opposite direction is applied injection: bottom up, from a finer graph to a coarser graph. Relaxation is an iterative process in which a graph's vertices are traversed, improving the solution. In order to increase the chances of a more accurate solution, the algorithm's direction is determined in a nondeterministic way using so-called Simulated Annealing. The MLAT implementation may enable a Multi-Processing leading to Parallel-Processing, which is an additional advantage.